My question refers to an article of Aaron Mazel-Gee about fibered categories in grupoids where the author introduces in a quite strange way a new category $\{X_0/X_1\}$ providing a functor $p: \{X_0/X_1\} \to C$ making $\{X_0/X_1\}$ into a category fibered in groupoids: https://etale.site/writing/stax-seminar-talk.pdf
We have following construction:
We start with a fixed category $C$ and consider the pair $(X_0,X_1)$ of two objects $X_1, X_1 \in C$ satisfying identities between maps $s, t , \epsilon, i $ and $m$ as described in the article. Here the excerpt:
Then comes the cruical point:
We take a $U \in C$ and define a category $\{X_0(U)/X_1(U)\}$.
The objects are defined via $ob(\{X_0(U)/X_1(U)\}):= X_0(U)$.
And exactly this is the problem: What is exactly $X_0(U)$? Especially how $X_0$ "acts" on $U \in C$.
Some days ago I asked a similar question and got two answers from @Victoria M and @Kevin Carlson. But up to now I'm quite not sure if I understood the explanations correctly. Here the link: Fibered Categories in Groupoids
Now I would like to try to explain how I understood it and I would be glad if anybody could look thought my interpretation attempts and take corrections if there is some point which I totally misunderstood:
In the setting as above the $X_i \in C$ themselves form a groupoid $(X_0,X_1)$ by definition iff they satisfy the identities between $s, t, \epsilon, i ,m$ (in sense of "internal groupoid").
For arbitrary $U \in C$ the only meaning for $X_i(U)$ seems logically to me comes with a "double interpretation" of the $X_i$:
as roughly elements of $C$ and as functors $X_i: C \to Grp$ given formally concretely via the $Hom$ functor $C(-, X_i)$.
Seems reasonable in light of Yoneda-lemma or more precisely Yoneda embedding. Then the expresion $X_0(U)$ is identified with $C(U, X_0)$. Seems plausible.
But is this interpretation of the category $X_0/X_1$ and $X_0(U)$ exacly that what the author meant or a failable attemp to find a interpretation?